Field Guide to
Optical Thin Films Ronald R.Willey
SPIE Field Guides Volume FG07 John E. Greivenkamp, Series Editor
SPIE
PRESS
Bellingham, Washington USA
Library of Congress Cataloging-in-Publication Data Willey, Ronald R., 1936Field guide to optical thin films I Ronald R. Willey. p. em. -- (SPIE field guides ; v. FG07) Includes bibliographical references and index. 1. Optical coatings. 2. Thin films. 3. Optical films. I. Title. II. Series: SPIE field guides; FG07. TS517.2.W535 2006 681'.4--dc22
2005037922
Published by SPIE-The International Society for Optical Engineering P.O. Box 10 Bellingham, Washington 98227-0010 USA Phone: + 1 360 676 3290 Fax: +1360 647 1445 Email:
[email protected] Web: http://spie.org
Copyright © 2006 The Society of Photo-Optical Instrumentation Engineers All rights reserved. No part of this publication may be reproduced or distributed in any form or by any means without written permission of the publisher. The content of this book reflects the work and thought of the author. Every effort has been made to publish reliable and accurate information herein, but the publisher is not responsible for the validity of the information or for any outcomes resulting from reliance thereon. Printed in the United States of America .
•
The International Society for Optical Engineering
Introduction to the Series
Welcome to the SPIE Field Guide~a senes of publications written directly for the practicing engineer or scientist. Many textbooks and professional reference books cover optical principles and techniques in depth. The aim of the SPIE Field Guides is to distill this information, providing readers with a handy desk or briefcase reference that provides basic, essential information about optical principles, techniques, or phenomena, including definitions and descriptions, key equations, illustrations, application examples, design considerations, and additional resources. A significant effort will be made to provide a consistent notation and style between volumes in the series. Each SPIE Field Guide addresses a major field of optical science and technology. The concept of these Field Guides is a format-intensive presentation based on figures and equations supplemented by concise explanations. In most cases, this modular approach places a single topic on a page, and provides full coverage of that topic on that page. Highlights, insights and rules of thumb are displayed in sidebars to the main text. The appendices at the end of each Field Guide provide additional information such as related material outside the main scope of the volume, key mathematical relationships and alternative methods. While complete in their coverage, the concise presentation may not be appropriate for those new to the field. The SPIE Field Guides are intended to be living documents. The modular page-based presentation format allows them to be easily updated and expanded. We are interested in your suggestions for new Field Guide topics as well as what material should be added to an individual volume to make these Field Guides more useful to you. Please contact us at John E. Greivenkamp, Series Editor Optical Sciences Center The University of Arizona
The Field Guide Series Keep information at your fingertips with all of the titles in the Field Guide Series:
Field Guide to Geometrical Optics, John E. Greivenkamp (FGOl) Field Guide to Atmospheric Optics, Larry C. Andrews (FG02) Field Guide to Adaptive Optics, Robert K. Tyson & Benjamin W. Frazier (FG03) Field Guide to Visual and Ophthalmic Optics, Jim Schwiegerling (FG04) Field Guide to Polarization, Edward Collett (FG05) Field Guide to Optical Lithography, Chris A. Mack (FG06) Field Guide to Optical Thin Films, Ronald R. Willey (FG07)
Field Guide to Optical Thin Films The principles of optical thin films are reviewed and applications shown of various useful graphical tools (or methods) for optical coating design: the reflectance diagram, admittance diagram, and triangle diagram. It is shown graphically how unavailable indices can be approximated by two available indices of higher and lower values than the one to be approximated. The basis of ideal antireflection coating design is shown empirically. The practical approximation of these inhomogeneous index profiles is demonstrated. Much of the discussions center on AR coatings, but most other coating types are seen in the perspective of the same graphics and underlying principles. is the basis of essentially all dielectric optical coatings; and transmittance, optical density, etc., are byproducts of reflection (and absorption). The best insight is gained by the study of reflectance. It is also shown that AR coatings, high reflectors, and edge filters are all in the same family of designs. The graphical tools described are found to be useful as an aid to understanding and insight with respect to how optical coatings function and how they might be designed to meet given requirements. Ronald R. Willey
Table of Contents Glossary
X
Fundamentals of Thin Film Optics Optical Basic Concepts Internal Angles in Thin Films Reflection Reflections Example Reflection Calculations
1 2 3 4 5
1
Graphics for Visualization of Coating Behavior Reflectance as Vector Addition Reflectance Amplitude Diagram Admittance Diagram Electric Field in a Coating Admittance versus Reflectance Amplitude Diagrams Triangle Diagram
6 6 7 8 9 10 11
Behavior of Some Simple AR Coating Types Single-Layer Antireflection Coating Two-Layer AR Amplitude Diagram Example Wavelength Effects Broad-Band AR Coating Two V-Coat Possibilities
12 12 13 14 17 18
Index of Refraction Simulations and Approximations Effective Index of Refraction Complex Effective Index Plot Simulating One Index with Two Others Herpin Equivalent Layers Approximations of One Index with Others
19 19 20 21 22 23
The QWOT Stack, a Coating Building Block QWOT Stack Reflectors QWOT Stack Properties Width of the Block Band Applications of the QWOT Stack Absentee Layer Narrow Band Pass Filter Vll
25 25 26 28 29 30 31
Table of Contents
Optical Density and Decibels (dB) NBP Filter Design Multiple-Cavity NBP Rabbit Ears
32 33 34 36
Coatings at Non-Normal Angles of Incidence Polarization Effects Wavelength Shift with Angle of Incidence Angle of Incidence Effects in Coatings Polarizing Beamsplitters Polarization as Viewed in Circle Diagrams Non-Polarizing Beamsplitters in General A Non-Polarizing Beamsplitter Design Procedure Non-Polarizing BS's Found & Rules-of-Thumb
37 37 38 39 40 41 43 44 46
Coatings with Absorption Various Metals on Triangle Diagrams Chromium Metal Details A Design Example Using Chromium Potential Transmittance
47 47 48 50 52
Understanding Behavior and Estimating a Coating's Potential 53 Estimating What Can Be Done Before Designing 53 Effects of Last Layer Index on BBAR Coatings 54 Effects of Index Difference (H- L) on BBAR Coatings 55 Bandwidth Effects on BBAR Coatings 56 Bandwidth Effects Background 57 Estimating the Rave of a BBAR 59 Estimating the Minimum Number of Layers in a 60 BBAR Bandpass and Blocker Coatings 61 Mirror Estimating Example Using ODBWP 63 Estimating Edge Steepness in Bandpass Filters 64 Estimating Bandwidths of Narrow Bandpass Filters 65 Blocking Bands at Higher Harmonics of a QWOT Stack 68
Vlll
Table of Contents Insight Gained from Hypothetical Cases "Step-Down" Index of Refraction AR Coatings Too Much Overall Thickness in a Design Inhomogeneous Index of Refraction Designs
71 71 74 75
Possibility of Synthesizing Designs Fourier Concepts Fourier Background Fourier Examples Fourier Limitations
77 77 78 80 82
Designing Various Types of Coatings Designing a New Coating Designing BEAR Coatings Tails in BEAR Coatings Designing Edge Filters, High Reflectors, Polarizing and Non-Polarizing Beamsplitters Designing Beamsplitters in General Designing to a Spectral Shape & Computer Optimization Performance Goals and Weightings Constraints Global vs. Local Minima Some Optimizing Concepts Damped Least Squares Optimization Needle Optimization Flip-Flop Optimization
84 84 85 87
Appendix Equation Summary
89 90 91 92 93 94 94 95 95 96 97 97
Bibliography Index
101 102
lX
Glossary
A Absorptance intensity absentee layer Layer of an even number of QWOTs thickness at the design wavelength angle matched PT of layer adjusted to be QWOT at an angle AOI Angle of incidence to surface normal Antireflection (coating) AR BBAR Broad band AR coating Beamsplitter BS Bandwidth BW Speed of light in vacuum c cavity A spacer of HWOT (or multiples) with high reflectors on either side cm- 1 Wavenumbers, number of waves per em. dB Decibel ( = -10 OD) DOE Design of experiments methodology DWDM Dense wavelength division multiplexing GHz Gigahertz, 109 cycles per second High-efficiency AR coating HEAR Half-wave optical thickness HWOT Imaginary, in complex numbers ~ Infrared IR Extinction coefficient k Layer pair (H and L) LP LWP Long wavelength pass (filter) MDM Metal-dielectric-metal coating Multiple internal reflection MIR Multilayer antireflection (coating) MLAR Matching layer(s) ML Real part of the index of refraction n Complex refractive index. N = n - ik N Effective index of refraction ne Narrow band pass (filter) NBP Optical density ( = log10 (1/T)) OD ODBWP Optical density bandwidth product OT Optical thickness PD Prism diagram PT Physical thickness
X
Glossary QHQ QWOT r
R Rabbit ears Rave
rugate SLAR SWP t T TIR
uv
v
wavenumber y
z a 8 A v ()
Quarter-half-quarter wave OT design of AR coating Quarter wave optical thickness Reflectance amplitude coefficient Reflectance intensity (R = rr*) High reflectance in the pass band of a NBP the spectral appearance is like rabbit ears Average reflectance over a spectral region Index versus thickness profile of rippled or corrugated form rather than square wave Single-layer antireflection (coating) Short wavelength pass (filter) Thickness Transmittance, intensity Total internal reflection Ultraviolet Speed of light in medium Number of wavelengths per centimeter in vacuum Admittance Optical axis Absorption coefficient Angle of incidence, refraction, or reflection Wavelength Frequency of the light Number of wavelengths per centimeter Phase
Xl
Fundamentals of Thin Film Optics
1
Optical Basic Concepts Optical thin films are primarily the study of interference between the multiple reflections of light from optical interfaces between two different media. Some principles and terminology are drawn from the basics of geometrical optics. Index of refraction n: n = clu,
u = c/n
c = 2.99792458xl0 8 m/s,
where c is the speed of light in vacuum and u is the speed of light in a medium. Wavelength A and frequency v :
"A=vlv,
in vacuum: A=c/v
The wave number o is the number of wavelengths per centimeter: o = 1/A units of cm-1• Optical path difference (OPD) is proportional to the time required for light to travel between two points. In homogeneous media: OPD = nd,
where d is the physical distance between the two points. Snell's law of refraction: n 1 sin 8 1 = n 2 sin 82 .
The incident ray, the refracted ray, and the surface normal are coplanar. When propagating through a series of parallel interfaces, the quantity n sin 8 is conserved.
Optical Thin Films
2
Internal Angles in Thin Films In a stack of plane parallel thin-film surfaces of various indices, the angle of a ray within a given film can be calculated using Snell's law. Because the law applies at each interface and follows from the previous interface, the internal angle in the film is independent of the order of the films or where that film is within the stack:
nz = 1.46
n1
=2.35
The immerging ray from the stack refracts back to an index of 1.0 and is parallel to the entering ray because the index is the same. Optical coatings on lenses are not strictly plane parallel thin-film surfaces, but on the scale of the thickness of these films(~ 1 ).liD), the approximation is valid. On a more microscopic scale (~ 1 nm), the surfaces of real coatings are not usually very smooth or flat. However, at wavelengths of one or two orders of magnitude greater than that, the approximation of "flat" is still valid. In the deep ultraviolet (UV) or x-ray region, these things can be of concern because of the shorter wavelengths.
Fundamentals of Thin Film Optics
3
Reflection Law of reflection :
The incident ray, the reflected ray, and the surface normal are coplanar. Total internal reflection (TIR) occurs when the angle of incidence of a ray propagating from a higher-index medium to a lower-index medium exceeds the critical angle :
At the critical angle, the angle of refraction 82 equals 90°. The reflectance amplitude r of an interface between n 1 and n 2 at normal incidence (8 1 = 0), with no absorption, is given by the Fresnel reflection equation:
When absorption is included:
where
When the angle of incidence (AOI) 8 1 is not 0°, the andp-polarizations have different effective indices: Ns
=n
X
cos 8 and N P
s-
= n I cos 8 ;
In these cases and Normally, thin-film calculation/design software takes care of all of these details.
Optical Thin Films
4
Reflections Reflectance intensity R is what is measured with a photometer. It is the product of the reflectance amplitude r and its complex conjugate r*:
R
= rr*.
Each interface between two media of differing index has Fresnel reflection. As seen in the figure, a ray falling on the first surface has r 1 reflected from it. The transmitted part has r 2 reflected when it falls on the second surface of the shaded medium. Part of this second reflection is reflected when it falls back on the first surface. Part of that reflection is reflected when it falls again on the second surface, etc.
When all of the reflections from the first and second surfaces of a thin film are considered, the resulting reflectance r is rigorously given by
Here, e-i cr is the complex phase factor that accounts for the phase difference between r 1 and r 2 caused by the optical thickness of the film: e-i cr
= cos qJ- i sin (jl.
These equations properly account for the multiple internal reflections in the medium.
Fundamentals of Thin Film Optics
5
Example Reflection Calculations A soap bubble might have an index of refraction of 1.4. The reflections in air (n = 1.0) from the first and second surfaces would be r 1 = (1- 1.4)/(1 + 1.4) = - 1/6. and r2 = (1.4- 1)/(1.4 + 1) = + 1/6. If the bubble is infinitesimally thin, the two reflections cancel each other: r1
where cp
+ r 2 = -1/6 + 1/6 = 0.0,
= 0, so that r
= (r 1 + r 2 *1)/(1 + r 1 r 2 *1) = 0.0 .
If the thin film of the bubble were one uu"~'"""" tJlic.kn,ess ( QWOT = 'A I 4 = nd ) at the wavelength under consideration, the path of the ray from the first surface to the second and back to the first would be one half wavelength. In this case cp = 180 deg. Here, the two reflections would add to each other for maximum effect:
r=[r1 +r2 *(-1)]/[1+r1 r2 *(-1)]=-0.3243, r""r1 +r2 ""-1/6+(-)1/6""-1/3.
The reflectance R would be rr* or -1/9, which is -11%. The appearance of the reflection when the bubble has a thickness of one QWOT would be white. As the bubble became thinner, its appearance would change toward no reflectance or "black." In the normal case, a bubble has a thickness of many QWOTs. For a given physical thickness, there are more QWOTs in blue light (short wavelength) than in the red (long wavelength). Therefore, the blue and red "rays" are at different phases and therefore have different reflectance. This causes the rainbow that we see in soap bubbles.
Optical Thin Films
6
Reflectance as Vector Addition The case of the soap bubble is illustrated from the viewpoint of vectors. When cp
= 0, r 1 = -1/6, and r 2 = + 1/6. Imaginary axis Real axis Zero reflectance point
/
,"
... ...
... r,
''
'
------,-~---+-------------Re I
'
r2
'.........
----
. ./
When cp = 180° (one QWOT):
I
'
,"
lm
... ...
...
r2
r1
''\
~-----+-------Re I \
Maximum reflectance point
When cp
= 360° (two QWOTs or one half-wave OT): lm
----~-~__~:~...~:~~j~------------Re
\-,......-...
Graphics for Visualization of Coating Behavior
7
Reflectance Amplitude Diagram The reflectance amplitude diagram , often referred to as a circle diagram , follows from the foregoing vector diagrams (p.6). lm
The outermost circle represents r = 1.0 and also R = rr* = 1.0 or R = 100% reflectance. The origin of the real and imaginary axes is where r = 0.0 or zero reflectance. Any new layer starts from the point of the reflectance amplitude of whatever lies beneath it, whether that is a substrate or a stack of coating layers on a substrate. This point is represented by point A in the figure, where the starting reflectance amplitude is rA and its phase is cpAAfter the addition of a given physical thickness (PT = d) and optical thickness (OT) of a layer of given index, the resulting reflectance reaches point B. The new reflectance amplitude is rB and its phase is cpB. Point B would then be the starting reflectance for the next layer.
Optical Thin Films
8
Admittance Diagram The admittance (Y) of a medium, as applied to optical thin films, is an electrical quantity, which is normalized to be equivalent to the index of refraction. When the effective index of a thin-film stack is plotted as a function of increasing thickness from the substrate to the end of the last layer, an admittance diagram is produced. This is a conformal mapping of the reflectance amplitude diagram. For many cases the diagrams look quite similar, except that they have been rotated 180° about a point. In the locus of homogeneous layers, nonabsorbing layers are circles. m +t2.0
+i1 .0
M
0,0
ZV""c
2.0
3.0 Re
L
-i1 .0
-i2 .0
The range of an admittance diagram is the semi-infinite plane from 0 to oo to the right on the real axis and ±ioo on the imaginary axis. The two-layer case of a QWOT of M and L is plotted here for comparison. Note that point S is at 1.52 -iO.O, the index of the substrate; C is at 1.90132 iO.O; and Z is at Y = Ne = 1.0 -iO.O, where r = (1.0 - 1.0) I (1.0 + 1.0 ) = 0.
Graphics for Visualization of Coating Behavior
9
Electric Field in a Coating The electric field within a coating layer is of importance when laser damage thresholds are considered and also when working with absorbing layers. In the latter case, the amount of energy absorbed in the layer depends upon the relative value of the electric field within that layer. One aspect of the laser damage issue seems to be that the interface between real deposited layers has some absorption and defects that are more vulnerable if the electric field is high at that interface. The relative volts/meter is a function of the real value of the admittance and can be calculated as follows: E
= 27.46 I [Re(Y)] 05 •
This is very convenient to visualize on an admittance diagram . The closer the locus of the coating gets to Y = 0, the higher the electric field, and therefore the more vulnerable the coating is to high energy flux. A laser stack is sometimes designed with non-QWOT layers that do not terminate toward the left on the real axis. Instead, the locus stops short of, or continues past, the crossing of the real axis until it is further away from the risks of high electric field at Y = 0 . 1·, 0
lm +i1 0
20
3 0 (Y)
i
E-field 30
20 V/M
605040
{"'\
0,0
z"" s) c "L
i1 .0
Re
Optical Thin Films
10
Admittance versus Reflectance Amplitude Diagrams
The admittance diagram is generally as useful as the reflectance amplitude or circle diagram , particularly in the realm of low reflectance (and thereby low admittance). However, the admittance diagram is not as useful for high reflectors such as a stack like (lH 1L)4, as can be seen in the figure below. For high reflectors of many layers, the admittance tends to go off of whatever scale is chosen for the plot, whereas the reflectance amplitude diagram is constrained to the unit circle. ADMITTANCE
S2
S6 20 YREAL
24
Reflectance Amplitude Diagram
-1
Graphics for Visualization of Coating Behavior
11
Triangle Diagram The energy falling on an optical thin film will be either reflected (R), transmitted (T), or absorbed (A). Scattering is ignored for the purposes of this section. It can then be stated that
R+T+A=l. A convenient way to visualize the properties of materials with absorption, such as metals, is using a triangle diagram. When there is no absorption (A), the transmittance (T) is simply 1 - R and the diagram is not particularly useful. This figure shows the path in R, T, and A for a coating of aluminum on glass. It starts at 96% T, 4% R, and 0% A on the bare glass surface. As the aluminum thickness increases, the locus moves downward. This happens to pass through a point of about 40% T, 43% R, and 17% A. When the film becomes opaque, it is at a point of 0% T, 90% R, and 10%A. These triangle diagrams are useful when a material has a significant value of k, the imaginary index, as in metals and semiconductors. 100%T
100%A
Optical Thin Films
12
Single-Layer Antireflection Coating The most common single-layer antireflection coating (SLAR) on glass is a QWOT of MgF 2 • If the index of the glass is 1.52 and that of the MgF 2 is 1.38:
11 = (1.0 -1.38)1(1.0 + 1.38) = -0.1597' r2 = (1.38 -1.52) I (1.38 + 1.52) = -0.0483.
When the film is infinitesimally thin,
q:J
= 0, so that
r = [(r1 +r2 *1)1(l+r1 r2 *I)]= -0.2064;
R = 4.260% (same as bare substrate). When the thin film is one QWOT, at the wavelength under consideration, then q:J = 180°:
J
J
r = [ r1 + r2 * ( -1) I [ 1 + r1r2 * ( -1) = -0.112 3 ;
R = 1.260%.
Thus, this SLAR reduces the reflection of the glass surface from 4.26% to 1.26% (only at the QWOT wavelength, and 0° AOI; at other wavelengths and angles, R is modified). The locus of this coating as seen on a reflectance amplitude or circle diagram as it grows from zero to one QWOT is a semicircle moving clockwise from point S to L.
13
Behavior of Some Simple AR Coating Types
Two-Layer AR Amplitude Diagram Example In this example (in air), the substrate is of index 1.52 and its reflectance amplitude point is at S. The first QWOT layer is of index M = 1. 70. The second QWOT layer is of index L = 1.38. This two-layer coating results in zero reflectance at the origin point, Z, for this design wavelength (only).
The supporting calculations are r:"ubstrate
= (1.0 -1.52)/(1.0 + 1.52) = -0.20635;Rs = 4.528%.
After the deposition of the first layer: r1 = (1.0 -1.70 )1(1.0 + 1. 70) = -0.25926 r2 =(1.70-1.52)1(1.70+1.52)=+0.05590;
<Jl=180°
rc = {[ -0.25926(-)( +0.05590) ]![1.0 + (-)- 0.25926 * (+0.05590) ]} = -0.310658; neffective = 1.90132;
Rc = 9.651%
After the deposition of the second layer: r1 = (1.0 -1.38)/ (1.0 + 1.38) = -0.15966
r2 = (1.38 -1.90132)1(1.38 + 1.90132) = -0.15888;<jl = 180°
rz
= {[ -0.15966(-)( -0.15888) =
-0.00080;
Rc
=
Jt[ (1.0 + (-)- 0.15966) * (-0.15888)]}
0.000064%!
Optical Thin Films
14
Wavelength Effects If the SLAR were designed to be just one QWOT at a 510nm wavelength, that same physical thickness (PT) would be 0. 729 QWOT at a wavelength of 700 nm, and 1.275 QWOT at 400 nm. This means that the semicircle shown previously (p. 12) would be less than a semicircle for 700 nm and more than one for 400 nm. The 400 and 700 nm terminations of the layer would be further from the origin than QWOT at 510 nm, and therefore would have a higher reflectance. Specifically, the reflectance would be 1. 799% at 400 nm and 1. 786% at 700 nm, as compared to 1.260% for 510 nm. lm
Re
s
The resulting reflectance versus wavelength would appear as the figure below. SLAR
2.0
15
1.0
""
........
--..
l.----
I"'"'"
--
v-
0.5
0
•oo
500
550
Wavelength {nm)
600
650
700
15
Behavior of Some Simple AR Coating Types
Wavelength Effects (cont.)
The two-layer AR design shown previously had one QWOT of index 1. 70 and one QWOT of index 1.38 to bring the TWOUmo... reflectance to essentially zero at the design wavelength. If this was designed J .. for 510 nm, both • layers would be \ 1.275x thicker at / ./ 400 nm and 0.729x thinner at 700 nm than one QWOT. The wavelength effects would be even more exaggerated than the SLAR case as shown.
\ \
It
.-
I
'
'
/
v
- ·--
Coatings such as these are referred to as "V-coatings" because of the V ,_u_.,. shape of the reflectance spectrum. However, if a • • layer of index 2.2 and thickness of two QWOTs were added between the two • .. layers, the result is shown. This additional half- wave lm OT layer is referred to as an achromatizing layer because it minimizes the Re changes in reflectance with color or wavelength. This might best be understood in the framework of the reflectance amplitude or circle diagrams at different colors/wavelengths. At the design wavelength of 510 nm, the first QWOT of index 1.7 moves the locus from the substrate to the left and on the negative real axis. The first QWOT of index 2.2 moves the locus from this point again to the left and on the negative real axis.
.
j'
\
\...
_,.... -Vi..
/
..
16
Optical Thin Films
Wavelength Effects (cont.) The second QWOT of index 2.2 moves clockwise back to the beginning of the 2.2 layer. The half-wave layer is called an absentee layer because it does not change the reflectance, and therefore acts as though it were absent. The last layer of index 1.38 moves the locus from this point to the origin of the coordinates, r = 0. lm
Re
At a wavelength of 650 nm, the layers are not as thick as a QWOT, and therefore are not full semicircles. All of the layers are too short, but the two QWOTs of the 2.2 index layer compensate for this by opening in such a way as to push the end point of the last layer toward the origin and closer tor = 0. At a wavelength of 450 nm the layers are thicker than a QWOT, and therefore are more than full semicircles on such a diagram. All of the layers are too long, but the two QWOTs of the 2.2 index layer compensate for this by closing in such a way as to pull the end point of the last layer toward the origin.
lm
Re
Behavior of Some Simple AR Coating Types
17
Broad-Band AR Coating The three-layer AR coating described on p. 15 is the basis of the broad-band AR (BEAR) coating, sometimes referred to as a high-efficiency AR (HEAR) coating. It has low reflection over a broad spectral range, particularly as compared to the V-coat. It has significantly lower reflection than the SLAR of MgF 2 on 1.52 index glass. It should be noted, however, that MgF 2 on a glass of index 1.90 (or near that) would have zero reflectance at the design wavelength and a moderately broad spectral AR band, as seen in the figure below. When high-index glasses receive an AR coating, a SLAR of MgF 2 may be a viable alternative to the BEAR. The three materials for the BEAR coating might be Al 2 0 3 for the medium index layer, Ti0 2 or Ta20 5 for the highindex layer, and MgF 2 for the low-index layer. The indices do not have to be exactly as illustrated here, and small adjustments from exact QWOTs can often balance the results in production for best spectral performance. This type of coating is probably the next most common production coating in the world, after the SLAR with MgF2 • SLAR MaF2 ON 1.90 INDEX GLASS
2.0
1.6
1.0
0.6
0 400
\
\
\
\ 450
~ r--... 600
/
560
Wavelength (nm)
/ 600
/
v
660
/ 700
Optical Thin Films
18
Two V-Coat Possibilities There are two possibilities for a V-coat with indices of 2.2 and 1.38. One (#1) is an H-layer with much less than a QWOT and an L-layer that is more than a QWOT.
(-----L-f--T----8-f----lm--:-.1
lm
.............. _____ , ........
The second (#2) has a # 2 first H-layer that starts out the same, but is much thicker before it Re terminates. A smaller Llayer com-pletes the AR coating.
However, #1 is less sensitive to changes of wavelength because it is a "flatter V." The first has a smaller locus due to fewer QWOTs of total thickness. As a result, it changes less as the wavelength changes. The broader design implies that production should be J .. ~~-\=:::j::=~-J.l~+-~ easier since it would be • more tolerant to small ~L-~---3~~ - ---L---L--~ changes of wavelength andAOI.
--
V-coat ARs can be designed for all wavelength regions as long as transparent materials are available for those regions. They can be designed from almost any ----"-----r-------Y-+---Re materials if they have enough difference in index of refraction. The example is in the infrared (IR) at 10.6 11m, where the substrate and the H-coating material is germanium of index 4.0, and L is a fluoride of index 1.35.
lm
Index of Refraction Simulations and Approximations
19
Effective Index of Refraction The last surface of any coating stack behaves as though it were a slab of homogeneous material of a specific refr~tct;ion neffectivel or ne, the value of which depends on all the layers of the stack and the substrate. It can be found from the last reflectance amplitude and phase computed for the coating stack. The final rand cp could be of any value from 0.0 to 1.0 and oo to 360°, respectively. The effective index can be found using the Fresnel reflection equation of a surface in air (or vacuum): r=(1-n)1(1+n).
Using simple algebra, this can be rearranged to give
n = ( 1- r) I (1 + r) = ne. When the phase of the final reflectance amplitude is on the real axis, so that cp is either oo or 180°, then this effective index is real and has no imaginary component. However, in the general case, r is a complex number and has an imaginary component. Therefore, ne is also a complex number and has an imaginary component. This means that it behaves like a general semiconductor or metallic material at that wavelength with absorption in terms of the resulting reflectance amplitude, which generates
However, if there have been no absorption losses in the coating stack, such losses are not implied. The reflection just looks this way, but the rest of the flux has been transmitted without absorption. Recall that R = rr*, where r* is the complex conjugate, so that the reflected intensity does not give evidence of the phase.
Optical Thin Films
20
Complex Effective Index Plot When the complex effective indices are calculated for general reflectance amplitudes and phases, we find ne and ke values as shown in the figure. This figure is symmetric about the horizontal (real) axis.
Note that r = 1.0 corresponds to an effective index of N. = 0.0 -iO.O, and r = -1.0 corresponds to an index of Ne oo -iO.O. Also, r = il.O corresponds to an index of Ne = 1.0 -il.O.
=
The following figure is the application of the above diagram to plot the n and k of the opaque point nickel (Ni) versus wavelength.
Index of Refraction Simulations and Approximations
21
Simulating One Index with Two Others It is possible, at one wavelength and AOI, to simulate a layer of any thickness and index by two or more layers of materials whose indices are greater and less than the index to be simulated. A typical example is the case of the threelayer BBAR discussed earlier. The materials used there were H = 2.2, lm M = 1.7, and L = 1.38. The M-layer can be Re replaced or simulated by a layer of H and L. These short H- and L- layers bring the locus to the same point as that of the replaced Mlayer. The reflectance versus wavelength is similar to the three-layer design. FOUR-LAYER TWO-MATERIAL AR
2.0
1.5
\
~
~
~
.."'
1.0
0.5
0
400
\ \
I
\ '-.... 450
/ 500
550
600
I
v
700
Wavelength (nm)
Note that a V-coat similar to the one previously shown will result if the second H-layer (two QWOTs) is eliminated. This would then be a two-layer AR with Hand L materials instead ofM and L. However, the layers are not QWOTs in the new case.
22
Optical Thin Films
Herpin Equivalent Layers
M. A. Herpin showed mathematically that any intermediate index layer could be simulated by a symmetric three-layer combination of HLH or LHL where the outer layers are of the same thickness. This gives a matrix representation that is the same, at one lm wavelength and AOI, as the layer being simulated or replaced. Although the Re first and third L-ayers appear to be of differing sizes, they do have the The LHL version with spectrum. same OT.
H
Re
The HLH version with spectrum.
HLH & LHL HERPIN REPLACEMENT IN 3·LAYER AR
2 » r------,----~------.------,------,------,
450
500
550
Wavelength (nm)
600
650
700
Index of Refraction Simulations and Approximations
23
Approximations of One Index with Others Herpin equivalent index approximation is a useful tool. Some more general cases are examined here. The admittance locus of a QWOT layer M of index 1.65 (such as in a three-ayer AR) that moves from a substrate point A of Y = 1.52 to a termination point at Z is illustrated. At this wavelength and angle (0°), any combination of layers that moves the locus from A to Z gives the same reflectance. The Herpin HLH travels the locus ABCZ, while the LHL solution travels the locus ADEZ. AFZ is another path that accomplishes the same thing.
ADMITTANCE
.5
...
D
.3
.2 .1
y IMAG.
0
- .1
- .2 E
- .3
f.O
1. 2
1.-
·...:
0
